Multivariable Model Predictive Control of Thin Film Surface Roughness and Slope for Light Trapping Optimization

Abstract

This work focuses on the development of a multivariable model predictive controller that simultaneously regulates thin film surface roughness and mean slope to optimize light reflectance and transmittance during thin film manufacturing by manipulating substrate temperature and deposition rate. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and the surface slope profile, respectively. The dynamics of the evolution of the thin film surface height profile are assumed to be described by an Edwards-Wilkinson-type equation (a second-order stochastic partial differential equation) in two spatial dimensions. Analytical solutions of the expected surface roughness and surface slope are obtained on the basis of the Edwards-Wilkinson equation and are used in the controller design. The model parameters of the Edwards-Wilkinson equation depend on the substrate temperature and deposition rate. This dependence is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope at the end of the growth process. The model predictive controller involves constraints on the magnitude and rate of change of the control action and optimizes a cost that involves penalty on both surface roughness and mean slope from the set-point values. The controller is applied to the two-dimensional Edwards-Wilkinson equation and is shown to successfully regulate surface roughness and mean slope to set-point values at the end of the deposition that yield desired film reflectance and transmittance.